On Lie Algebras of Generalized Jacobi Matrices

Home , Carl Gustav Jacob Jacobi

**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2020

ON LIE ALGEBRAS OF GENERALIZED JACOBI MATRICES

ALICE FIALOWSKI University of Pécsand EötvösLorándUniversity Budapest, Hungary E-mail: fi[email protected], fi[email protected]

KENJI IOHARA Univ Lyon, Universit´eClaude Bernard Lyon 1 CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France E-mail: [email protected]

Abstract. In these lecture notes, we consider inﬁnite dimensional Lie algebras of generalized

Jacobi matrices gJ(k) and gl∞(k), which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra gJ(k) and of the orthogonal and symplectic subalgebras.

1. Introduction. In this expository article, we consider a special type of general linear Lie algebras of infinite rank over a field k of characteristic zero, and compute their homology with trivial coefficients. Our interest in this topic came from an old short note (2 pages long) of Boris Feigin and Boris Tsygan (1983, [FT]). They stated some results on the homology with trivial coefficients of the Lie algebra of generalized Jacobi matrices over a field of characteristic 0. It was a real effort to piece out the precise statements in that densely encoded note, not to speak about the few line proofs. In the process of understanding the statements and proofs, we got involved in the topic, found different generalizations and figured out correct approaches. We should also mention that their old results generated much interest during these 36 years. In the meantime, two big problems have beed solved, which helped

2010 Mathematics Subject Classification: Primary 17B65, 16S35; Secondary 16E40. Key words and phrases: Infinite dimensional Lie algebras, Lie algebra Homology, Cyclic and Hochschild Homology, smash product, spectral sequence. The paper is in final form and no version of it will be published elsewhere.

[1] 2 A. FIALOWSKI AND K. IOHARA us to understand the statements and work out the right proofs. One is the Loday-Quillen- Tsygan theorem (see (3.1)) and the other is the generalization of the Hochschild-Serre type spectral sequence by Stefan (see (3.4)), without which the statements of the old note could not be justified. Consider the Lie algebra gJ(k) of generalized Jacobi matrices, namely, infinite size matrices M = (mi,j) indexed over Z such that mi,j = 0 if |i − j| > N for some N de- pending on the matrix M. The original Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix (a band matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.) It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel mea- sure. This operator is named after Carl Gustav Jacob Jacobi, who introduced in 1848 tridiagonal matrices and proved the following Theorem: every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix (see [K] and e.g. [Sz]). Since then, Jacobi matrices play an important role in different branches of mathematics, like in topology (Bott’s periodicity theorem on homotopy groups), stable homotopy theory, algebraic geometry and C∗-algebras (see [Ka1]). They are also used to show some interesting properties in K-theory. For instance, Karoubi used them to prove the conjecture of Atiyah-Singer about the classifying space, see [Ka2].

The Lie algebra gl∞(k) of finitely supported infinite size matrices indexed over Z can be naturally viewed as a subalgebra of gJ(k). The Lie algebra gJ(k) has typical infinite dimensional nature. For example, the two matrices X X P := iEi−1,i,Q := Ei+1,i, i∈Z i∈Z where Ei,j is the matrix unit with 1 on the (i, j)-entry, belong to gJ(k) but not to gl∞(k). They satisfy 1. tQ · Q = I, 2. PQ − QP = I, P where I denotes the identity matrix Ei,i ∈ gJ(k). This matrix I does not even i∈Z d belong to gl (k) ! (Off course, the matrices P and Q are matrix representations of ∞ dt ±1 i and t· on the vector space C[t ] with respect to the basis {t }i∈Z.) Such algebras show up in many areas of mathematics and physics. For instance, they are used to describe the solitons of the Kadomtsev-Petviashvili (KP) type hierarchies [Sa] where such integrable systems are interpreted as a dynamical system on the so-called Sato Grassmannian. On the other hand, their basic algebraic properties and invariants are not well understood. In our work we present results on their homology with trivial coefficients, For this, we need to use several different (co)homology concepts. Some results were obtained by Feigin and Tsygan [FT] in 1983, but in their short note the statements and proofs are not precise. In our work, we were able to get straightforward statements and proofs and we also could generalize the results to the coefficients over an associative unital k-algebra. GENERALIZED JACOBI MATRICES 3

The structure of the paper is as follows. In Section 2 we introduce some important classes of Lie algebras of general Jacobi matrices and recall their universal central extension. We also give some examples of its subalgebras. Section 3 is devoted to their (co)homology. First we recall the main definitions: Lie algebra homology, Hochschild homology, cyclic homology and (skew)dihedral homology. Then we compute homology with trivial coefficients for the introduced Lie algebras. In this section, we also present precise proofs of some results in [FT] and give possible generalizations. In Section 4 we introduce two important subalgebras, the orthogonal and symplectic subalgebras. To compute their (co)homology, we need to introduce additional computational methods to the previous one. In Section 5 we discuss a more general case, where instead of the field k we have an associative unital k-algebra, and generalize our (co)homology results for such algebras. Finally, we introduce a rank functional on a Lie subalgebra gJ∞(k) ⊆ gJ(k) and describe its image. After describing its cohomology ring, we also raise some open questions.

2. Lie algebras of generalized Jacobi matrices.

2.1. Lie algebras gl∞(k) and gJ(k). The first such Lie algebra one may have in mind is the one defined as an inductive limit: let I be a countable set and I1 ( I2 ( ··· ( In ( ··· , S I = n In an increasing series where each In is a finite subset. The Lie algebra glI (k) is defined by the inductive limit of glIm (k) ,→ glIn (k) for m < n, namely, each element of glI (k) is a finitely supported matrix of infinite size indexed over I, i.e., X = (xi,j)i,j∈I 2 such that ]{(i, j) ∈ I | xi,j 6= 0 } < ∞. In particular, for I = Z, we may denote by gl∞(k). Another one we may also encounter is the Lie algebra of generalized Jacobi matrices defined as follows. A generalized Jacobi matrix is a matrix M = (mi,j) indexed over Z such that there exists a positive integer NM satisfying

mi,j = 0 ∀ i, j such that |i − j| > NM . The set of such matrices has a structure of associative algebra over k denoted by J(k). We shall denote it by gJ(k) whenever we regard it as Lie algebra. An original Jacobi matrix is a ﬁnite size matrix M = (mi,j) such that mi,j = 0 for any i, j with |i − j| > 1. The Lie algebra gl∞(k) can be naturally viewed as a subalgebra of gJ(k). 2.2. Universal central extension of the Lie algebra gJ(k). In the course of study- ing soliton theory, a non-trivial central extension of the Lie algebra gJ(k) was discovered (see, e.g., [JM] and [DJM]) and it can be described as follows. P Let J = i≥0 Ei,i ∈ gJ(k) be a matrix and let Φ : J(k) → J(k) be the k-linear map deﬁned by X 7→ JXJ. It can be checked that, for any X,Y ∈ J(k), the element

[Φ(X), Φ(Y )] − Φ([X,Y ]) is an element of gl∞(k), i.e., only finitely many matrix entires can be non-zero. Hence, one can define the k-bilinear map Ψ : J(k) × J(k) → k by Ψ(X,Y ) = tr([Φ(X), Φ(Y )] − Φ([X,Y ])), P where tr : gl (k) → k ; X = (xi,j) 7→ xi,i is the trace of finitely supported matrices. ∞ i∈Z It turned out that this Ψ is a 2-cocycle, called Japanese cocycle, i.e., 1.Ψ( Y,X) = −Ψ(X,Y ), 4 A. FIALOWSKI AND K. IOHARA

2. Ψ([X,Y ],Z) + Ψ([Y,Z],X) + Ψ([Z,X],Y ) = 0, for any X,Y,Z ∈ J(k). Let gfJ(k) := gJ(k) ⊕ k · 1 be the Lie algebra whose Lie bracket [·, ·]0 is given by [X, 1]0 = 0, [X,Y ]0 = [X,Y ] + Ψ(X,Y )1 X,Y ∈ gJ(k). As the Lie algebra gJ(k) is perfect, i.e., [gJ(k), gJ(k)] = gJ(k), the Lie algebra gJ(k) admits the universal central extension. It was B. L. Feigin and B. L. Tsygan [FT] in 1983 who proved that this central extension α : gfJ(k) → gJ(k) is universal, namely, gfJ(k) is perfect and for any central extension β : a → gJ(k), there exists a morphism of Lie algebras γ : gfJ(k) → a such that the next diagram commutes: α gfJ(k) / gJ(k) A

γ β a Remark 2.1. The kernel z of the universal central extension of gJ(k), i.e., the kernel of the canonical projection gfJ(k) gJ(k) can be given by the 2nd homology H2(gJ(k)) and the 2-cocycle Ψ is an element of the 2nd cohomology H2(gJ(k)).

Remark 2.2. Let GfJ be the (pro-)algebraic group of gfJ(k). The Lie algebra gfJ(k) acts on a fermionic Fock space F. The GfJ -orbit of its vacuum state | vaci in the projective space PF is isomorphic to the Sato Grassmannian. The deﬁning equations of this orbit in terms of Pl¨ucker coordinates is nothing but the Hirota bilinear equations of Kadomtsev- Petviashvili (KP) hierarchy. For details, see, e.g., [JM] and [DJM].

2.3. Some subalgebras of the universal central extension gfJ(k). The extended Lie algebra contains several interesting inﬁnite dimensional Lie algebras as subalgebras. We shall introduce some of them.

1. For an integer n > 1, let gJn(k) be the subalgebra of gfJ(k) generated by the matrices M = (mi,j)i,j∈Z satisfying mi+n,j+n = mi,j for any i, j ∈ Z. This subalgebra is −1 isomorphic to the central extension of gln(k[t, t ]), i.e., the aﬃne Lie algebra −1 glcn(k) = gln(k[t, t ]) ⊕ kc whose commutation relation is given by a b a+b [ei,j ⊗ t , ek,l ⊗ t ] = [ei,j, ek,l] ⊗ t + aδa+b,0δj,kδi,lc, [glcn(k), c] = 0, where ei,j ∈ gln is the matrix unit whose (i, j)-entry is 1. An isomorphism between glfn(k) a P and gJn(k) is given by ei,j ⊗ t 7→ e . g r∈Z i+rn,j+(r+a)n

2. Another important example is the one-dimensional central extension of the Lie d algebra k[t, t−1] of algebraic differential operators over k∗ = k \{0} that is defined dt d as follows. Set D = t dt . For any polynomial f, g ∈ k[D], it can be verified that [trf(D), tsg(D)] = tr+s(f(D + s)g(D) − f(D)g(D + r)). GENERALIZED JACOBI MATRICES 5

d Let Ψ be the 2-cocyle on k[t, t−1] deﬁned by dt ( P f(j)g(j + r) r = −s ≥ 0, Ψ(trf(D), tsg(D)) := −r≤j≤−1 0 r + s 6= 0. d The Lie algebra W is, by deﬁnition, the central extension of k[t, t−1] by the 1+∞ dt 2-cocycle Ψ, i.e., it is the k-vector space d W = k[t, t−1] ⊕ kC, 1+∞ dt equipped with the Lie bracket given by r s 0 r s r s [t f(D), t g(D)] = [t f(D), t g(D)] + Ψ(t f(D), t g(D))C, [W1+∞,C] = 0.

It can be shown that there exists morphism of Lie algebras W1+∞ ,→ gfJ(k) satisfying a r P a −1 t D 7→ i Ei+r,i. We remark that the Lie subalgebra k[t, t ]D ⊕ kC of W1+∞ is i∈Z isomorphic to the Virasoro algebra. The ﬁrst example shows that the Lie algebra gfJ(k) contains, at least, aﬃne Lie (1) (1) (1) (1) (2) algebras of classical type, i.e., Al (l ≥ 1),Bl (l ≥ 3),Cl (l ≥ 2),Dl (l ≥ 4),A2l (l ≥ (2) (2) (3) 1),A2l−1, (l ≥ 3),Dl+1(l ≥ 2) and D4 . In addition, the second example shows that Lie algebra gfJ(k) contains also the Lie algebra W1+∞ that plays an important role in the KP-hierarchy.

3. Homology of the Lie algebra gJ(k). In this Section, among others we state the main result of [FT] and explain the outline of the proof. Our may goal is the computation of the homology H•(gJ(k)). 3.1. Several homologies. We brieﬂy recall the deﬁnitions of Lie algebra (co)homology, Hochschild homology, cyclic homology and (skew-)dihedral homology.

Let g be a Lie algebra over a field k of characteristic 0. From now on, we shall abbre- viate the coefficient k. The Lie algebra homology H•(g) is, by definition, the homology of the complex (V• g, d), called the Eilenberg-Chevalley complex, where V• g is the exterior algebra of g and the differential d is given by X i+j+1 d(x1 ∧ · · · ∧ xn) = (−1) [xi, xj] ∧ x1 ∧ · · · xî ∧ · · · ∧ xˆj ∧ · · · ∧ xn. 1≤i

∆(x) = x ⊗ 1 + 1 ⊗ x. For the Lie algebra gl∞(R) over an associative unital k-algebra R, the primitive part of the homology H•(gl∞(R)) had been known by Loday and Quillen [LQ] and independently by B. L. Tsygan [Ts] as follows:

Theorem 3.1. The primitive part Prim H•(gl∞(R)) is isomorphic to the cyclic homology HC•−1(R). The cohomology ring H•(g) is naturally endowed with a commutative and cocommu- n ∼ ∗ tative DG-Hopf algebra structure. Indeed, by the Poincar´eduality H (g) = Hn(g) (full • ∼ ∨ L ∗ dual), one has H (g) = H•(g) := n Hn(g) (restricted dual). Now we recall some deﬁnitions in homology theory of associative algebras. For detail, see, e.g., [Lod].

Let R be an associative unital k-algebra and M be an R-bimodule. For n ∈ Z≥0, ⊗n set Cn(R,M) = M ⊗ R . The Hochschild homology H•(R,M) is, by definition, the homology of the complex (C•(R,M), b), where the the differential b is defined by

b(m ⊗ r1 ⊗ · · · ⊗ rn) n−1 X i =ma1 ⊗ a2 ⊗ · · · ⊗ an + (−1) m ⊗ a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an i=1 n + (−1) anm ⊗ a1 ⊗ · · · ⊗ an−1.

In particular, for M = R, for each n ∈ Z>0, there is an action of the cyclic group Z/(n + 1)Z given by n x.(r0 ⊗ r1 ⊗ · · · ⊗ rn) = (−1) (rn ⊗ r0 ⊗ · · · ⊗ rn−1), where x is a generator of the group Z/(n + 1)Z. The diﬀerential b of the complex λ ⊗(n+1) (C•(R,R), b) induces a diﬀerential on the complex Cn (R) := R /(1 − x). The homology of this complex, called Connes’ complex is the so-called cyclic homology HC•(R) of R. For some cases, this cyclic homology can be computed with the aid of Connes’ periodicity exact sequence:

· · · → HHn(R) → HCn(R) → HCn−2(R) → HHn−1(R) → · · · . (1) Now, assume that R is equipped with a k-linear anti-involution ¯· : R → R. One can ⊗(n+1) extend the action of the group Z/(n + 1)Z on R to the dihedral group Dn+1 = hx, y|xn+1 = y2 = 1, yxy = x−1i by 1 n(n+1) y.(r0 ⊗ r1 ⊗ · · · ⊗ rn) = (−1) 2 (r0 ⊗ rn ⊗ rn−1 ⊗ · · · ⊗ r1). ⊗n+1 Let Dn(R) denote the space of coinvariants (R )Dn+1 . If we modify the action of y by −y, the resulting coinvariants will be denoted by −1Dn(R). The diﬀerential b on ⊗n+1 ¯ C•(R,R) induces the diﬀerentials on (R )Dn+1 and −1Dn(R), denoted by b. Their homologies are called dihedral (resp. skew-dihedral) homology of R: ¯ ¯ HDn(R) := Hn(D•(R), b), ( resp. −1HDn(R) := Hn(−1D•(R), b)). For more informations, see, e.g., [Lod]. GENERALIZED JACOBI MATRICES 7

∼ 3.2. An isomorphism gJ(k) = gln(J(k)). Let n > 1 be an integer. We ﬁx a section of

Z Z/nZ and denote its image by I. For any matrix M = (mi,j)i,j∈Z and any i, j ∈ I, we set Mi,j := (mi+nk,j+nl)k,l∈Z. Then the k-linear map

ΦI : gJ(k) −→ gln(J(k)); M 7−→ (Mi,j)i,j∈I , is an isomorphism of Lie algebras gJ(k) −→ gln(J(k)), and it induces an isomorphism of ∼ homologies H•(gJ(k)) = H•(gln(J(k))). Taking an inductive limit, we obtain ∼ Lemma 3.2 (Lemma 1 of [FT]). H•(gJ(k)) = H•(gl∞(J(k))).

Hence, the homology H•(gJ(k)) is the commutative and cocommutative DG-Hopf algebra whose primitive part is HC•−1(J(k)) by Theorem 3.1. Thus, if we can compute the Hochschild homology HH•(J(k)) of the associative algebra J(k), Connes’ periodicity exact sequence (1) allows us to determine the cyclic homology HC•(J(k)). In the rest of this section, we will explain this brieﬂy.

3.3. Twisted Group Algebras. Let A be an associative unital k-algebra and let G be a discrete subgroup of the group of k-automorphisms of A. One can twist the natural product structure on A ⊗k k[G] by (a ⊗ [g]) · (b ⊗ [h]) := ag(b) ⊗ [gh], where a, b ∈ A and g, h ∈ G. The tensor product A ⊗k k[G] equipped with such a twisted product is called twisted group algebra and will be denoted by A{G}.

n z }| { Example 3.3. 1. Fix n ∈ Z>1. Let A = k × k × · · · × k be the n-copies of k viewed as commutative associative algebra. The group G = Z/nZ acts on A via cyclic permutation: (i + nZ).(a1, . . . , an) = (ai+1, . . . , ai+n) where the indicies should be understood modulo n. Then, the twisted group algebra A{G} is isomorphic to the algebra of n × n matrices Mn(k). The isomorphism A{G} → Mn(k) is given by n X (a1, . . . , an) ⊗ [i + nZ] 7−→ akek,k+i. k=1 Q 2. A = kei, where ei’s are orthogonal idempotents and G = . The additive i∈Z Z group Z acts on A: 1.ei = ei−1. One can show that the twisted group algebra A{G} is isomorphic to the associative algebra J(k) where the isomorphism A{G} → J(k) is given by X X akek ⊗ [i] 7−→ akEk,k+i. k∈Z k∈Z 3.4. Hochschild-Serre type Spectral Sequence. Let M be an A{G}-bimodule.

Theorem 3.4. [St] There exists a spectral sequence

2 Ep,q = Hp(k[G],Hq(A, M)) =⇒ Hp+q(A{G},M) Remark 3.5. In [FT], B. Feigin and B. Tsygan treated the above case when M = A{G}. But their description of the k[G]-module structure on Hq(A, A{G}) was not correct and this was rectiﬁed by D. Stefan [St] under more general setting. 8 A. FIALOWSKI AND K. IOHARA

In our case, G = Z and k[Z] is a principal ideal domain, hence its global dimension is at most 1 which implies that this spectral sequence collapses at E2.

By a standard argument, one can show that ∼ Y Hq(A, J(k))) = HHq(k)ei. i∈Z Moreover, by direct computation, it can be verified that ( ∼ HHq(k) p = 1, Hp(k[Z],Hq(A, J(k))) = 0 p 6= 1. ∼ This implies HHp(J(k)) = HHp−1(k). ∼ Remark 3.6. The isomorphism H1(k[Z],Hp(A, J(k)) = HHp+1(J(k)) is given by the so-called shuffle product [Q]. This fact plays an important role when we determined the homology of the Lie algebras of orthogonal and symplectic generalized Jacobi matrices in [FI2]. By definition, it follows that ( k p = 0, HHp(k) = 0 p > 0. Thus, we obtain

Theorem 3.7 (cf. Theorem 3 in [FT]). HH1(J(k)) = k and HHp(J(k)) = 0 for any p 6= 1. By Connes’ Periodicity long exact sequence (1), one has

Corollary 3.8. HCp(J(k)) = k for odd p and HCp(J(k)) = 0 for even p.

3.5. Description of the primitive part Prim (H•(gJ(k))). Theorem 3.1 and Corol- lary 3.8 imply

Theorem 3.9. Prim (H•(gJ(k))) is the graded k-vector space whose pth graded component is ( k p ≡ 0(2), Prim (H•(gJ(k)))p = 0 otherwise.

In particular, each graded subspace of H•(gJ(k)) is of ﬁnite dimension. Thus, by the p ∼ ∗ Poincar´eduality H (gJ(k)) = Hp(gJ(k)) , one obtains Theorem 1. a) in [FT], i.e., the cohomology ring H•(gJ(k)) has the next description: 2i • Theorem 3.10. There exists primitive elements ci ∈ H (gJ(k)) such that H (gJ(k)) is L isomorphic to the Hopf algebra S( kci). i∈Z>0

4. Orthogonal and Symplectic Subalgebras of gJ(k). In this Section, after recall- ing the deﬁnition of orthogonal and symplectic subalgebras of gJ(k), we present two key steps to compute their homology of these subalgebras and state the results about their homology. GENERALIZED JACOBI MATRICES 9

4.1. Deﬁnitions. Let R be an associative unital k-algebra equipped with an k-linear anti-involution ¯·, i.e., s¯ = s and st = t¯s¯ for s, t ∈ R. We extend the transpose, also denoted t t by (·), to J(R) by (Ei,j(r)) = Ej,i(¯r), where Ei,j(r) = Ei,jr ∈ J(R). s For l ∈ Z, let τl, τl be the k-linear anti-involutions of the Lie algebra gJ(k) deﬁned by l t X i τl(X) = (−1) Jl( X)Jl,Jl := (−1) Ei,l−i, i∈Z s s t s s X τl (X) = Jl ( X)Jl ,Jl := Ei,l−i. l∈Z We set

s odd τ0 ,− s oJ (k) =gJ(k) = {X ∈ gJ(k)|τ0 (X) = −X},

τ−1,− spJ (k) =gJ(k) = {X ∈ gJ(k)|τ−1(X) = −X}, s even τ−1,− s oJ (k) =gJ(k) = {X ∈ gJ(k)|τ−1(X) = −X}. odd Remark 4.1. The universal central extension of the Lie algebras gJ(k), oJ (k), spJ (k) even and oJ (k) are the Lie algebras of type AJ ,BJ ,CJ and DJ respectively that are used to obtain the Hirota bilinear forms with these symmetries. See, e.g., [JM], for details.

4.2. Stable limit of Orthogonal and Symplectic Lie algebras and homology. Set X X i X JB = ei,−i,JC = (−1) ei,−i−1,JD = ei,−i−1. i∈Z i∈Z i∈Z

We deﬁne the anti-involutions τB, τC and τD of gl∞(R) as follows: t t t τB(X) = JB( X)JB, τC (X) = −JC ( X)JC , τD(X) = JD( X)JD. The Lie algebras

τB ,− τC ,− τD ,− oodd (R) := gl∞(R) , sp(R) := gl∞(R) , oeven (R) := gl∞(R) are clearly the stable limit of the Lie algebras {o2l+1(R)}l∈Z>0 , {sp2l(R)}, {o2l(R)}l∈Z>1 , respectively. In a way similar to the case gJ(k) (cf. Lemma 3.2), one can show Lemma 4.2. There exists isomorphisms of homologies: odd ∼ 1. H•(oJ (k)) = H•(oodd (J(k))), ∼ 2. H•(spJ (k)) = H•(sp(J(k))), even ∼ 3. H•(oJ (k)) = H•(oeven (J(k))), Thanks to Theorem 5.5 in [LP], due to J. L. Loday and C. Procesi, we obtain Theorem 4.3. Let ∗ : J(k) → J(k) be the anti-involution satisfying ∗ Er,s(a) = E−s,−r(a) a ∈ k. odd even Then, for g = oJ , spJ and oJ , we have

Prim (H•(g(k))) = −1HD•−1(J(k)), where −1HD•(·) signiﬁes the skew-dihedral homology. 10 A. FIALOWSKI AND K. IOHARA

It is slightly technical, but one can show (cf. [FI2]) that

−1HD•−1(J(k)) = HD•−2(k).

Notice that HDp(k) = k for p ≡ 0 mod 4 and HDp(k) = 0 otherwise. Thus, we have odd even Theorem 4.4 (cf. [FI2]). For g = oJ , spJ and oJ , Prim (H•(g(k))) is the graded k-vector space whose pth graded component is ( k p ≡ 2(4), Prim (H•(g(k)))p = 0 otherwise.

odd Remark 4.5. By Theorems 3.9 and 4.4, the support of Prim (H•(g(k))) for g = gJ, oJ , even spJ and oJ , i.e., those integers p where Prim (H•(g(k)))p 6= 0, are exactly the twice of the exponents of the Weyl group of type AJ ,BJ ,CJ and DJ , respectively.

5. Generalizations and some further topics. Here, we give a generalization on the coeﬃcient ring, mention some other related topics and open questions. 5.1. Lie algebras with coeﬃcients in R. The computations of Lie algebra homology given in Sections 2 and 3 can be generalized to the Lie algebra g(R), where g = odd even gJ, oJ , spJ and oJ . Here R is an associative unital k-algebra (equipped with a k- odd even linear anti-involution ¯· : R → R for g = oJ , spJ and oJ ). The next theorem generalizes Theorem 3.9:

Theorem 5.1 (cf. [FI1]). 1. Prim (H•(gJ(R))) = HC•−2(R), ∼ 2. H•(gJ(R)) = S(HC•−2(R)). The proof goes as follows. As in the case R = k, By Theorem 3.1, the primitive part Prim (H•(gJ(R))) is given by HC•−1(J(R)). Now, D. Stefan’s spectral sequence ∼ allows us to show an isomorphism HH•(J(R)) = HH•−1(R). Analyzing carefully this ∼ isomorphism, it can be shown that HC•−1(J(R)) = HC•−2(R). The above proof also allows us to obtain a generalization of Theorem 4.4: odd even Theorem 5.2 (cf. [FI2]). Set g = oJ , spJ and oJ .

1. Prim (H•(g(k))) = HD•−2(R), ∼ 2. H•(g(R)) = S(HD•−2(R)).

5.2. Rank and trace functionals. For any n ∈ Z>1, let gJn(k) be the image of the composition gJn(k) ,→ gfJ(k) gJ(k). Let gJ∞(k) be the subalgebra of gJ(k) generated by all of the gJn(k)(n ∈ Z>1). Viewed as an associative algebra, gJn(k) and gJ∞(k) will be denoted by Jn(k) and J∞(k), respectively. We don’t know whether this algebra J∞(k) is a von Neumann regular ring, i.e., for any A ∈ J∞(k), there exists X ∈ J∞(k) such that AXA = A. Nevertheless, there is a so-called rank functional deﬁned as follows.

For A = (ai,j)i,j∈Z ∈ J∞(k), set 1 Rank(A) = lim rank(An), n→∞ 2n + 1 where, the matrix An of size 2n + 1 is defined by (ai,j)−n≤i,j≤n and rank( · ) signifies the rank of a finite size matrix. GENERALIZED JACOBI MATRICES 11

Remark 5.3. It was pointed out by B. Feigin and B. Tsygan in [FT] that one can deﬁne the trace functional Tr : J∞(k) → k by 1 Tr(A) = lim tr(An), n→∞ 2n + 1 where tr(·) is the trace of a ﬁnite size matrix. They used this functional to describe non • trivial cocycles of the cohomology H (gJ∞(k)).

It turns out that Im(Rank) = [0, 1]∩Q. Indeed, this rank functional is a rank function on J∞(k) in the sense of J. von Neumann [vN], i.e., it satisﬁes, 1. Rank(I) = 1, 2. Rank(xy) ≤ Rank(x), Rank(y), 3. Rank(e+f) = Rank(e)+Rank(f) for all orthogonal idempotents e, f ∈ J∞(k), and 4. Rank(x) = 0 if and only if x = 0.

Hence, let Jc∞(k) ⊂ J(k) be the Rank-completion of J∞(k). We denote the continuous extension of Rank( · ) to Jc∞(k) by Rank( · ) (cf. [G]). Proposition 5.4. Im(Rank) = [0, 1]. Proof. It suﬃces to show that for any x ∈ [0, 1] ∩ Qc, there exists a diagonal D = diag(di)i∈Z such that Rank(D) = x.

Let {rn}n∈Z≥0 ⊂ Z>0 be the sequence defined as follows. Set r0 = 1. For n > 0, let rn rn−1 rn be the integer such that 2n+1 < x < 2n+1 . Such an integer exists since x is irrational. It 1 can be checked that if 0 < x < 2 , then rn+1 − rn ∈ {0, 1}, otherwise, rn+1 − rn ∈ {1, 2}. Now, we define the sequence {di}i∈Z. Set d0 = 1. Suppose that {di}|i|≤n is defined. Choose d±(n+1) ∈ {0, 1} in such a way that ]{i ∈ {±(n + 1)}|di = 1} = rn+1 − rn. For any such choice, it follows that the rank of the diagonal matrix diag(di)|i|≤n is rn for any n ∈ Z≥0. Hence, we have Rank(D) = x by construction. There might be a von Neumann regular subalgebra of J(k) that has not been explored up to now. Here are some questions:

1. Does Jc∞(k) admit a Lie algebra structure ? If yes,

i) and if the Lie algebra gJc∞(k) is perfect, describe its universal central extension. ii) More generally, can we have a description of the (co)homology ring of Lie algebra gJc∞(k)?

2. Is J∞(k) von Neumann regular algebra ? If yes, so is Jc∞(k) (cf. [G]). i) What kind of additional properties, do they have ?

5.3. More about gJ∞(k). In [FT], B. Feigin and B. Tsygan determined the structure of the cohomology ring H•(gJ(k)) which states, for each i > 0, there exists, up to scalar, 2i unique ci ∈ H (gJ(k)) such that • ∼ • M H (gJ(k)) = S ( kci). i>0 12 A. FIALOWSKI AND K. IOHARA

This follows from Theorem 3.9. The natural inclusion gJ∞(k) ,→ gJ(k) induces a surjec- • • tive map H (gJ∞(k)) H (gJ(k)). Indeed, the showed that, for each i > 0, there exists, 2i−1 up to scalar, unique ξi ∈ H (gJ∞(k)) such that the next short sequence is exact: ^ • M • • 0 −→ ( kξi) −→ H (gJ∞(k)) −→ H (gJ(k)) −→ 0. i>0 They even presented an explicit realization of cocycles. It may be an interesting problem to obtain such descriptions also for the Lie algebras odd even gJ∞ (k) := gJ (k) ∩ gJ∞(k), where gJ = oJ , spJ and oJ . Acknowledgments. The authors thank Max Karoubi and Andrey Lazarev for helpful discussions. Kenji Iohara would like to thank the organizers for giving him the opportunity to present some of the results in this article.

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